IMO 1982 LL USS51

Let n numbers x1, x2, . . . , xn be chosen in such a way that

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IMO 1982 LL USS51

Origin: USS

Problem

Let n numbers x1, x2, . . . , xn be chosen in such a way that 1 \geqx1 \geqx2 \geq\cdot \cdot \cdot \geqxn \geq0. Prove that (1 + x1 + x2 + \cdot \cdot \cdot + xn)\alpha \leq1 + x\alpha 1 + 2\alpha−1x\alpha 2 + \cdot \cdot \cdot + n\alpha−1x\alpha n if 0 \leq\alpha \leq1.